You have correctly identified the major component of seashells as
calcium carbonate, CaCO3. Unfortunately there is not a lot of
chemistry of which to take advantage. However, we can learn from the
popular Easter craft of
coloring eggs, whose shells are also calcium carbonate. Those shells are
porous which aids in the coloring process, but the technique is the same.
You can use a commerical egg-coloring kit, or simply use food coloring and
follow the instructions on the box for coloring eggs. The trick is to add
a little vinegar to the hot water so as to "etch" the surface of the shell
so that the coloring can penetrate the crystalline structure. White
shells work best.

Like all good chemists, I did the experiment myself. I had best
results with red coloring and weathered shells (naturally a little porous
due to the aging process). You can see the result below compared to a
similar, untinted, shell on the right.

You might need to clean
your shells prior to staining them. A coat of clear lacquer or
varnish will protect the finished item.

You could also try a different type of ink-like dye
which claims to provide a waterproof color on almost any type of
surface.